LONDON/PARIS (Reuters) - France's presidential elections has become so unpredictable that gamblers are betting on a second term for the sitting president, even though he bowed out of the contest last year and his party has since picked another candidate.

British bookmakers reopened the market for bets on the deeply unpopular Socialist President Francois Hollande due to demand from customers.

The chances of him changing his mind and going on to win are now put at 100-to-1, where a 1 pound ($1.25) bet would pay out 100 pounds.

"People are much more willing to put a few pounds on the outsider at the moment," said Naomi Totten of the British betting site Betfair, which has taken 3 million pounds in French election bets to date.

"There are certainly people interested in backing him (Hollande) despite the fact he has very clearly ruled himself out."

Betting on politics is illegal in France, but the country's online gaming authority says people will not be prosecuted for betting abroad.

"We’ve picked up a lot of interest from (French) expats who can have a bet here but can't have a bet in Paris," said Graham Sharpe of William Hill, a British bookmaker.

Uncertainty is the lifeblood of the gambling industry, and the French presidential election has plenty of that.

Of the four leading candidates, two - conservative Francois Fillon and Socialist Benoit Hamon - were not in the picture three months ago. A third, Emmanuel Macron, was an outside bet.

Now Macron is the frontrunner for both the pollsters and the bookmakers, the result of a slump in Fillon's ratings after a scandal erupted two weeks ago over taxpayers' money funneled to his wife.

And in an election race full of surprises, it remains unknown what inroads Hamon, now fourth in the polls, might yet make, having won the Socialist Part nomination only a week ago.

"In the last year we've had four different people being favorite ... so that's good for us," said Matthew Shaddick, head of political odds at Ladbrokes.

10,000 POUND BET ON LE PEN

The one steady runner in the contest has been the leader of France's far-right National Front, Marine Le Pen.

Voter surveys have consistently shown the anti-EU, anti-immigration Le Pen reaching the May 7 second round runoff between the two top scorers from the first round a fortnight earlier.

They also show her losing that head-to-head vote by a big margin, with whoever faces her picking up at least 60 percent of the vote.

Nevertheless, betting has been heaviest on Le Pen, and the odds are short, at 5-to-2, where a 2 pound bet wins just 5 pounds.

About 43 percent of William Hill's takings on the French election to date are on Le Pen, Sharpe said, adding that on Friday one punter was unusually bold.

"We took the biggest single bet we've ever taken for a foreign election, other than the British or American ones," he said. "Ten thousand pounds for Marine Le Pen to win, from someone who walked into one of our south London betting shops."